diff --git a/source/precalculus/source/06-TR/03.ptx b/source/precalculus/source/06-TR/03.ptx index 8668d118..6bf39f0b 100644 --- a/source/precalculus/source/06-TR/03.ptx +++ b/source/precalculus/source/06-TR/03.ptx @@ -17,16 +17,22 @@ In this section, we will learn how to use right triangles to evaluate trigonometric functions. Before doing that, however, let's review some key concepts of right triangles that can also come in handy when solving.

+

- The Pythagorean Theorem is a^2+b^2=c^2, where a and b are lengths of the legs of a right triangle and c is the length of the hypotenuse. In other words, the theorem tells us that if we know the lengths of any two sides of a right triangle, we can find out the length of the third side. + The Pythagorean Theorem is a^2+b^2=c^2, where a and b are lengths of the legs of a right triangle and c is the length of the hypotenuse. If we know the lengths of any two sides of a right triangle, we can find the length of the third side.

- - Suppose the hypotenuse of a right triangle is 13 cm long and one of the legs is 5 cm long. + + +

+ Suppose two legs of a right triangle measure 3 inches and 4 inches. +

+ +

@@ -34,14 +40,54 @@

+

+ Students should draw a triangle with x representing the hypotenuse and the two legs 3 and 4 inches respectively. +

+ + + + +

+ What is the value of x (i.e., the length of the third side)? +

    +
  1. 5 inches
    2. +
  2. \sqrt{5} inches
    3. +
  3. 25 inches
    4. +
  4. 16 inches
+

+ + +

+ A +

+ + + + + + +

+ Suppose the hypotenuse of a right triangle is 13 cm long and one of the legs is 5 cm long. +

+ + + + +

+ Draw a picture of this right triangle and label the sides. Use x to refer to the missing side. +

+ + +

- One example of how a student can draw the triangle. + One example of how a student can draw the triangle.

+

@@ -61,14 +107,18 @@ - - The Pythagorean Theorem can be very helpful in finding the third side of a right triangle as long as we know the length of two other sides. What if we are given the side length of only one side of a given right triangle? How could we find the lengths of the other two sides? - + + +

+ Pythagorean triples are integers a, b, and c that satisfy the Pythagorean theorem. and highlight some of the most common types of Pythagorean triples: 3-4-5 and 5-12-13. All triangles similar to the 3-4-5 triangle will also have side lengths that are multiples of 3-4-5 (like 6-8-10). Similarly, this is true for all triangles similar to the 5-12-13 triangle. +

+ +

- Before we can answer the question posed, we must first make sure we know how to call specific sides of a right triangle. + When working with right triangles, it is often helpful to refer to specific angles and sides. One way this is done is by using letters, such as A and a to show that these are an angle-side pair because every angle has a side opposite the angle in a triangle. Note that the capital letter indicates the angle, and the lower case letter indicates the side. Another way to label the sides of a triangle is to use the relationships between a given angle within a triangle and the sides.

  • @@ -78,7 +128,7 @@

  • - The opposite side is across from a given angle. + The opposite side is the non-hypotenuse side across from a given angle.

  • @@ -87,22 +137,486 @@

-

When given an angle, all sides of a triangle can be labeled. For example, suppose angle A is given, then the sides of a right triangle can be labeled as:

- From the perspective of angle A, all sides of a right triangle can be labeled. + From the perspective of angle A, all sides of a right triangle can be labeled.

- + + +

+ Suppose you are given a right triangle where the hypotenuse is 11 cm long and and one of the interior angles is 60 degrees. +

+ + + + +

+ Draw a picture of this right triangle and label the sides (refer back to to help you label the sides). +

+ + + + + +

+ What is the measure of the third angle? +

    +
  1. 90 degrees
    2. +
  2. 60 degrees
    3. +
  3. 30 degrees
    4. +
  4. 180 degrees
+

+ + +

+ C +

+ + + + + +

+ Suppose you are asked to find one of the sides of the right triangle. What additional information would you need to find the length of another side of the triangle? +

+ + +

+ Students will probably notice that the Pythagorean Theorem is not helpful in this case because they only know the length of one side. This is a great opportunity to discuss how the Pythagorean Theorem is useful in finding side lengths when at least two sides are known. +

+ + + + + +

+ The Pythagorean Theorem can be very helpful in finding the third side of a right triangle as long as we know the length of two other sides. In , only one side and one angle were given. In this case, the Pythagorean Theorem is not enough to help us find another side of the right triangle (unless it is one of the triples!). +

+ + + + +

+ Trigonometric ratios such as sine, cosine, and tangent are based on the relationships between a given angle \theta and side lengths in a right triangle. Three trigonometric functions, sine, cosine, and tangent, are often used to understand the relationships between a given angle of a triangle and its sides. For acute angles, such as \theta, these functions can be defined as ratios between the sides of a right triangle. + + + \sin\theta=\frac{opposite}{hypotenuse} + + + \cos\theta=\frac{adjacent}{hypotenuse} + + + \tan\theta=\frac{opposite}{adjacent} + + +

+

+ Notice that these are defined according to the sides of a triangle - which is why it is important to be able to label correctly! +

+ + + + + +

The top of the Eiffel Tower is seen from a distance of d = 500 meters at an angle of \alpha=31 degrees. +

+ + + + +

+ Draw a diagram to represent the situation. Use x to refer to the missing side. +

+ + +

+ Students should draw a right triangle where \theta is the angle formed with the ground, 500 meters as the side adjacent to \theta, and the height of the Eiffel Tower (side opposite \theta) as x. +

+ + + + + +

+ Which trig function could we use to find the height of the tower? +

    +
  1. sine
    2. +
  2. cosine
    3. +
  3. tangent

+ + +

+ C +

+ + + + + +

+ How could we correctly set up the trigonometric ratio to find the height of the Eiffel Tower? +

    +
  1. \sin{31}=\frac{x}{500}
    2. +
  2. \cos{31}=\frac{500}{x}
    3. +
  3. \cos{31}=\frac{500}{x}
    4. +
  4. \tan{31}=\frac{x}{500}

+ + +

+ D +

+ + + + + +

+ Find the height of the tower. +

    +
  1. 428.58 meters
    2. +
  2. 300.43 meters
    3. +
  3. 220.85 meters
    4. +
  4. 257.52 meters

+ + +

+ B +

+ + + + + + +

+ For each triangle given, determine which trigonometric ratio would be the most helpful in determining the length of the side of a triangle. Be sure to draw a picture of the triangle to help you determine the relationship between the given angle and sides. +

+ + + + +

+ In triangle ABC, B=37 degrees and a=11. Which trigonometric function will best help determine the length of side c? +

    +
  1. sine
    2. +
  2. cosine
    3. +
  3. tangent

+ + +

+ B +

+ + + + + +

+ In triangle ABC, A=32 degrees and b=13. Which trigonometric function will best help determine the length of side a? +

    +
  1. sine
    2. +
  2. cosine
    3. +
  3. tangent

+ + +

+ C +

+ + + + + +

+ In triangle ABC, angle A=24 degrees and the hypotenuse is 14. Which trigonometric function will best help determine the length of side a? +

    +
  1. sine
    2. +
  2. cosine
    3. +
  3. tangent

+ + +

+ A +

+ + + + + + +

+ Suppose you are given triangle ABC, where a=35, b=12, and c=37, with c being the hypotenuse of the triangle. +

+ + + + +

+ Find the ratio of \tan{B}. +

    +
  1. \frac{35}{12}
    2. +
  2. \frac{35}{37}
    3. +
  3. \frac{12}{35}
    4. +
  4. \frac{12}{37}

+ + +

+ C +

+ + + + + +

+ Find the ratio of \cos{A}. +

    +
  1. \frac{35}{12}
    2. +
  2. \frac{35}{37}
    3. +
  3. \frac{12}{35}
    4. +
  4. \frac{12}{37}

+ + +

+ D +

+ + + + + +

+ Find the ratio of \sin{B}. +

    +
  1. \frac{35}{12}
    2. +
  2. \frac{35}{37}
    3. +
  3. \frac{12}{35}
    4. +
  4. \frac{12}{37}

+ + +

+ D +

+ + + + + +

+ Suppose we want to know the measure of angle A. We can find the measure of angle A in three different ways by using either sine, cosine, or tangent (since all side lengths are given). For each trigonometric function, write the trigonometric ratio that can be used to find the measure of angle A. +

+ + +

+ Students should be able to write all three trigonometric functions: \cos{A}=\frac{12}{37}, \sin{A}=\frac{35}{37}, and \tan{A}=\frac{35}{312}. +

+ + + + + +

+ Now that we have set up a trigonometric ratio to help find the measure of angle A, how can we use these ratios to determine how big A is? +

+ + +

+ Give students the opportunity to discuss with one another on how they would try to determine the measure of angle A. Instructors might want to give a hint about how to "undo" the trigonometric function. +

+ + + + + +

Sometimes you will need to use trigonometric functions to find the measure of an angle. In these cases, you will need to use the inverse trig function key on your calculator (such as \sin^(-1))to find the angle that yields that trig value. +

+

For example, the sine function \sin takes an angle and gives us the ratio \frac{opposite}{hypotenuse}, but \sin^{-1}(called "inverse sine") takes the ratio \frac{opposite}{hypotenuse} and gives us an angle. +

+ + + + +

+ Refer back to , where you were given all the sides of a right triangle, but no angle measures. (In triangle ABC, a=35, b=12, and c=37, with c being the hypotenuse of the triangle). +

+ + + + +

+ What is the trigonometric ratio for \cos{A}? +

    +
  1. \frac{35}{12}
    2. +
  2. \frac{35}{37}
    3. +
  3. \frac{12}{35}
    4. +
  4. \frac{12}{37}

+ + +

+ D +

+ + + + + +

+ Use the inverse trig function, \cos^{-1} to find the measure of angle A. (Make sure your calculator is in degree mode!) +

+ + +

+ Students should get approximately 71.08 degrees. +

+ + + + + +

+ What is the trigonometric ratio for \sin{A}? +

    +
  1. \frac{35}{12}
    2. +
  2. \frac{35}{37}
    3. +
  3. \frac{12}{35}
    4. +
  4. \frac{12}{37}

+ + +

+ B +

+ + + + + +

+ Use the inverse trig function, \sin^{-1} to find the measure of angle A. (Make sure your calculator is in degree mode!) +

+ + +

+ Students should get approximately 71.08 degrees. +

+ + + + + +

+ What is the trigonometric ratio for \tan{A}? +

    +
  1. \frac{35}{12}
    2. +
  2. \frac{35}{37}
    3. +
  3. \frac{12}{35}
    4. +
  4. \frac{12}{37}

+ + +

+ A +

+ + + + + +

+ Use the inverse trig function, \tan^{-1} to find the measure of angle A. (Make sure your calculator is in degree mode!) +

+ + +

+ Students should get approximately 71.08 degrees. +

+ + + + + +

+ Refer back to parts (b), (d), and (f). What do you notice about your answers from those parts? +

+ +

- Right triangles can be used to evaluate trigonometric functions, such as sine and cosine. Let's explore how to evaluate all six trigonometric functions! + Students should notice that they got the same angle measure for A regardless of which trigonometric function they used.

- + + + + + +

+ Now that we know the measure of angle A, find the measure of angle B. +

+ + +

+ Angle B is approximately 18.92 degrees. +

+ + + + + +

+ Determining all of the side lengths and angle measures of a right triangle is known as solving a right triangle. In and , we were given all the sides of the triangle and used trigonometric ratios to determine the measure of the angles. +

+ + + + +

+ Solve the following triangles using your knowledge of right triangles, the Pythagorean Theorem and trigonometric functions. Be sure to draw a picture to help you determine the relationship between the angles and sides. +

+ + + + +

+ In triangle ABC, B=53 degrees and c=5 meters (with c being the hypotenuse). +

+ + +

+ A=37 degrees, C=90 degrees, a=3 meters, and b=4 meters. +

+ + + + + +

+ In triangle ABC, A=28 degrees and b=29.3 miles (with c being the hypotenuse). +

+ + +

+ B=62 degrees, C=90 degrees, a=15.6 miles, and c=33.2 miles. +

+ + + + + +

+ In triangle ABC, a=8 feet, b=17 feet, and c=15 feet (with b being the hypotenuse). +

+ + +

+ A=28.07 degrees, B=90 degrees, and C=61.93 degrees. +

+ + + + diff --git a/source/precalculus/source/06-TR/04.ptx b/source/precalculus/source/06-TR/04.ptx index 49fb1b60..2e5fbecd 100644 --- a/source/precalculus/source/06-TR/04.ptx +++ b/source/precalculus/source/06-TR/04.ptx @@ -17,6 +17,7 @@ Recall from the previous section that we can find values of trigonometric functions of right triangles. In this section, we can take what we know about right triangles and use the same idea to find exact values of trigonometric functions of special angles.

+

@@ -45,7 +46,7 @@

- Draw a right triangle and label the angles to have 45, 45 , and 90. + Draw a right triangle and label the angles to have 45, 45, and 90.

@@ -127,14 +128,14 @@

Unlike the 45-45-90 triangle, we cannot easily find another side length with the information given. Using trigonometry, which trigonometric function would help us find the length of the hypotenuse with the information given?

    -
  1. \sin$\theta$
    2. -
  2. \cos$\theta$
    3. -
  3. \tan$\theta$
+
  • \sin\theta
    • +
  • \cos\theta
    • +
  • \tan\theta

    • - \sin$\theta$ + A